I’m lucky enough to be doing some work in Primary this term. It’s a great experience and in an ideal world this would be something all Secondary Maths teachers get the opportunity to do at some time. I’m working with Years 3-6 and have had some really interesting small group sessions with the children looking at multiplication.

In Year 3, we progressed from multiples of 10 (e.g. 2 x 30, 6 x 60) one day to full two digit multiplication (e.g. 5 x 24) the next day. I felt like this was a fairly rapid progression for Year 3. Although some of them coped fine, it was a lot for some. Later I got chance to do the same with some low attaining Year 5 students. Maybe unsurprisingly, the methods hadn’t stuck. Unsurprising because I have taught plenty of Year 7s who would struggle with this.

## 60 x 4

First, let’s start with multiplying multiples of 10. The standard way of teaching this to Year 3 seemed to result in this:

Steps are:

- Cross out the zero
- Do the multiplication of the other digits
- Then multiply that by 10 to get the answer.

Or, in some cases this was shortened to:

- Cross out the zero
- Do the multiplication
- Put the zero back.

The problem I had with this is that the end result didn’t make sense, i.e. what does this mean?So I tried to encourage them to just “cross out the zero in your head”. It felt awkward to me as I was saying it!

We are trying to get a balance here between developing a deep understanding of what is going on and learning an efficient method. I think we want students to move quite quickly to doing this mentally. After all, if you know 6×8, you know 6×80 and once you understand what is going on, more writing (i.e. writing the steps above) doesn’t really help. I think some Diennes blocks would really help at this point with the understanding.

Or you could try virtual manipulatives here or here.

Personally, I would have moved onto hundreds next (e.g. 3 x 500) to show that this is a general principle before attempting full 2 digit multiplication and spent some more time on that before moving onto the Box Method.

## 24 x 3 – The Box Method (also called Grid Method)

So, next day, next piece of learning and we now attempt two digit multiplications using the box method. At secondary we always try to wean students off the box method in Year 7 and get them doing formal long division, i.e. this.I was really pleased to see that the Year 5 students that I helped were already using the formal method. However, I have worked with some Year 10s that still use it. When you are doing big numbers you get some fairly ridiculous column addition at the end and many more *Opportunities for Error*.

Having said that, let’s not forget that it can be a useful as *Another Way to Show* algebraic multiplication, and some students find it useful for multiplication with decimals.

The key piece of understanding at the beginning is that we are breaking down the 2-digit number into its tens and its ones, i.e. 24 can be broken down into 20 and 4 and the distributive law of multiplication allows us to multiply each part separately. So the children are taught to fill in the squares in a box, i.e. this.Now I have no real evidence for the next statement, but personally I don’t see the point in the box method. Why not write out the two multiplications? i.e. this:It’s not much more writing and I think it is clearer. Specific problems I saw with the Box Method included:

- Drawing the boxes in the first place. Fine if someone does it for you, but when that scaffolding is removed…
- Remembering which numbers go where when setting it up
- Understanding the array structure, i.e. multiply the number at the top of the column by the number at the left of the row.
- Remembering which numbers to then use for your column addition.

So, why do we teach the Box Method? Especially as we then ditch it as soon as possible to teach formal long multiplication. I’m not sure why. My son in Year 6 isn’t sure why. I’d love to hear your thoughts.

The only advantage I can see is an alternate way to show the partial products method to scaffold to the usual long algorithm. I personally show this using the distributive property (which is probably what you envision when you say ‘just write out the multiplications’). Personally I like the box method if it is explicitly tied back to the partial products method. But they way you have above doesn’t seem to make that connection, so makes it feel a bit disconnected to me.

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Hi Casper! Thanks for making me read this again! Always good to revisit posts from a while ago. That sounds like a good progression to me. I’ve only recently started teaching the Distributive Law as a “thing” (probably needs a separate blog post!) but think it is really powerful to able to refer back to when teaching arithmetic and algebra.

In terms of methods, it’s a common situation, right? : finding the balance between conceptual understanding and an efficient method that “works” as in it yields a correct solution must of the time.

What I was thinking here is, what’s the benefit of drawing the grid vs just doing the calculations when doing e.g 34×5. And then maybe 35×15.? If we are trying to reinforce understanding, better to write out all the calculations than use the “method” of boxes. I agree, the diagonal boxes is different again and often gets even more fraught.

Final thought, why type of students are doing box method in Yr10 vs those doing formal long multiplication?

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Hey Mark! Thanks for this. I was planning on teaching the box method in my school because it naturally builds on prior lessons on (mentally) multiplying integers by 10, 100, 1000. I can use these previous lessons and the distributive law to explain WHY the method works. I imagine finding it much harder to explain how the formal method works. Agree it’s the trickiest method but isn’t that a price worth paying for better mathematical understanding? (Also, a separate point – isn’t the method with the diagonal lines separate from the box method?)

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